A short proof of w₁^n(Hom(C_(2r+1), K_(n+2)))=0 for all n and a graph colouring theorem by Babson and Kozlov

We show that the n-th power of the first Stiefel-Whitney class of the Z_2-operation on the graph complex Hom(C_{2r+1},K_{n+2})$ is zero, confirming a conjecture by Babson and Kozlov. This proves the strong form of their graph colouring theorem, which they had only proven for odd n. Our proof is also considerably simpler than their proof of the weak form of the theorem, which is also known as the Lov\'asz conjecture.